Is Every Open Neighborhood of a Limit Point in a T_1 Space Infinitely Populated?

[Bashyboy]

Homework Statement

Problem: Let $A$ be an infinite subset of a $T_1$ space, and let $x$ be a limit point of $A$. Prove that every open neighborhood of $x$ contains infinitely many points of $A$.

Homework Equations

The Attempt at a Solution

First note that if $\mathcal{O}$ is an open neighborhood of $x$ not containing $y$, then there exists open neighborhoods $U$ and $V$ containing $x$ and $y$, respectively, but not containing the other point. Hence, $\mathcal{O} \cap U \subseteq \mathcal{O}$ is an open set containing $x$ but not containing $y$.

Proof of main theorem: Let $\mathcal{O}$ be an open neighborhood of $x$. Then there exists $a_1 \in \mathcal{O} \cap (A-\{x\})$. By the observation made above, we can find an open neighborhood $\mathcal{O}_2$ of $x$ that doesn't contain $a_1$ and $\mathcal{O}_1 \subseteq \mathcal{O}$. Hence, there exists $a_2 \in \mathcal{O}_2 \cap (A-\{x,a_1\})$. Hence $(a_n)$ is an infinite sequence of distinct points in $A$ such that $a_n$ is in $\mathcal{O}$ for every $n \in \mathbb{N}$, thereby proving that $\mathcal{O}$ contains an infinite number of points in $A ~ \Box$.How does this sound? It seems little shaky; perhaps it could be made slightly more rigorous.

 

[RUber]

It sounds like you are assuming that there are an infinite number of points, and showing that there are an infinite number of subsets.

 

[RUber]

I would recommend starting with the definitions of T_1 space, infinite subset, and limit point. Use your definitions as fuel for your proof.